Method for controlling an actuator in a nested friction mechanical system

ABSTRACT

A method of controlling an electrical actuator of a mechanical system having a plurality of nested zones of contact, the method comprising the steps of:
         acquiring data about the mechanical system, which system includes a number of nested zones of contact;   preparing a model of the system on the basis of said data and of a number of LuGre models put in parallel equal to the number of nested zones of contact, and determining parameters of the model and also a compensation structure for compensating friction in the nested zones of contact;   including the compensation structure in a control relationship for the actuator A; and   controlling the actuator by means of the control relationship.

The present invention relates to the field of controlling electricalactuators in mechanical systems that include a movable element that ismovable relative to a stationary element.

BACKGROUND OF THE INVENTION

Numerous mechanical systems incorporate one or more electricalactuators, such as electromechanical actuators, that are connected to anelectrical power supply and that are controlled to move the movableelement of the mechanical system relative to the stationary element in amanner that is very accurate. Each actuator is controlled on the basisof a control relationship, e.g. associating parameters of the electricalpower supplied to the actuator with positions or speeds of the movableelement.

When two parts having friction surfaces in contact that rub against eachother when the movable element moves, it is necessary to control theactuator so that it produces sufficient force to overcome such frictionand so that the movement takes place without jerking. For this purpose,the control relationship includes a compensation structure enabling theactuator to be powered so that it overcomes friction and moves themovable element in a manner that is smooth.

There are several models in existence that enable friction to bemodelled. These models are based on the behavior of roughnesses presenton the contacting services, and the roughnesses are modeled in the formof spring blades in such a manner that the spring blades of one of thecontacting surfaces interact with rigid blades of the other contactingsurface.

The LuGre model can be used to model friction when only two parts are incontact, and it enables a stable friction compensation structure to bederived therefrom, as described in the document by De Wit et al., “A newmodel for control of systems with friction”, pages 419 to 425, No. 3,volume 40, IEEE Transactions on automatic control, 1995, IEEE. FIG. 1shows a part P resting on a surface S and moving at a speed v withfriction F opposing the movement of the part P, and also shows itsrepresentation in accordance with the LuGre model. The LuGre model isthen written in the following form:

$\left\{ \begin{matrix}{{\Gamma_{f}(t)} = {{\sigma_{0}{z(t)}} + {\sigma_{1}{\overset{.}{z}(t)}} + {\sigma_{2}{\omega(t)}}}} \\{{\overset{.}{z}(t)} = {{\omega(t)} = {\frac{❘{\omega(t)}❘}{g\left( {\omega(t)} \right)}{z(t)}}}} \\{{g(\omega)} = \frac{\Gamma_{c} + {\left( {\Gamma_{s} - \Gamma_{c}} \right)e^{- {(\frac{\omega(t)}{\omega_{s}})}^{2}}}}{\sigma_{0}}}\end{matrix} \right.$where:

-   -   Γ_(f)(t) is the total friction torque;    -   σ₀ is the stiffness of the spring blades;    -   σ₁ is the damping coefficient of the movement of the spring        blades;    -   σ₂ is the viscous friction coefficient;    -   Γ_(c) is the Coulomb torque;    -   Γ_(s) is the Stribeck or static friction torque;    -   ω_(s) is the Stribeck angular velocity; and    -   ω is the relative angular velocity.

There exist other friction models that are more accurate than the LuGremodel, but that are more complex in their structure (and are thus morecomplex to implement), such as the de Leuven model (described in thedocument by Lampaert et al., “Modification of the Leuven integratedfriction model structure”, pages 683 to 687, No. 4, volume 47, IEEEtransactions on Automatic Control, 2002, IEEE) or the GeneralizedMaxwell Slip (GMS) model (Lampaert et al., “A generalized Maxwell-slipfriction model appropriate for control purposes”, pages 1170 to 1177,volume 4, 2003 IEEE International Workshop on Workload Characterization,2003, IEEE). As a result of their complexity, those models can also beused to model mechanical architectures that are more complex. This isparticularly true when a plurality of contacting surfaces are nested inone another, as happens with a shaft pivoting in a first bushing that isitself pivoting in a second bushing that in turn is pivoting in abearing, possibly together with a sealing gasket rubbing against one oranother of the moving parts: it is not possible to determine theindividual contributions of the various contacting surfaces in theoverall friction that exists between the shaft and the bearing.

Nevertheless, those models do not enable the friction that they model tobe compensated in a manner that is stable.

OBJECT OF THE INVENTION

A particular object of the invention is to propose means for improvingthe accuracy and the precision of control over an actuator of amechanical system having a plurality of contacting surfaces.

SUMMARY OF THE INVENTION

To this end, according invention, there is provided a method ofcontrolling an electrical actuator of a mechanical system having aplurality of nested zones of contact, the method comprising the stepsof:

-   -   acquiring data about the mechanical system, which system        includes a number of nested zones of contact;    -   preparing a model of the system on the basis of said data and of        a number of LuGre models equal to the number of nested zones of        contact, and determining parameters of the model and also a        compensation structure for compensating friction in the nested        zones of contact;    -   including the compensation structure in a control relationship        for the actuator; and    -   controlling the actuator by means of the control relationship.

The method of the invention makes it possible to model any type ofmultiple friction and to compensate effectively the friction as modelledin this way so as to obtain a control relationship that is particularlyaccurate and stabilizing.

Other characteristics and advantages of the invention appear on readingthe following description of a particular and nonlimiting implementationof the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

Reference is made to the accompanying drawings, in which:

FIG. 1 is a representation of a friction situation and of the way it ismodelled by the conventional LuGre model;

FIG. 2 is a diagrammatic view showing a rotating mechanical system inwhich the method of the invention can be implemented;

FIG. 3 is a diagrammatic view of the control device;

FIG. 4 is a diagrammatic representation of the compensator of thecontrol device; and

FIG. 5 is a diagram of a servo control loop used for implementing themethod of the invention.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 2 , the mechanical system used for illustrating the operation ofthe invention comprises four parts in relative friction, e.g. such as arotor of an electric motor turning in a stator and constrained to rotatewith a driven shaft that turns in a bearing for guiding rotation. Insuch an arrangement, there are at least three friction zones that arecumulative and that can be represented by four parts 1, 2, 3, and 4 thatare coaxial one within another and that are rotating relative to oneanother, the four parts being in contact with one another as follows:

-   -   the part 1 has an outside surface S1′ that is in contact with an        inside surface S2 of the part 2;    -   the part 2 has an outside surface S2′ that is in contact with an        inside surface S3 of the part 3; and    -   the part 3 has an outside surface S3′ that is in contact with        the inside surface S4 of the part 4.

The system thus has three friction zones: S1′/S2, S2′/S3, and S3′/S4.These friction zones are said to be nested.

The method of the invention is arranged to control the electric motor,which constitutes the electrical actuator of the mechanical system. Thecontrol method comprises the steps of:

-   -   acquiring data about the mechanical system that comprises a        number M of nested zones of contact;    -   preparing a model of the system, referred to as a M-LuGre model,        on the basis of said data and of a number M of LuGre models put        in parallel, where M is equal to the number of nested zones of        contact, with this being done by using a stochastic optimization        algorithm in order to determine parameters of the model and in        order to determine a compensation structure for compensating        friction in the nested zones of contact;    -   including the compensation structure in a control relationship        for the actuator; and    -   controlling the actuator by means of the control relationship.

The M-LuGre model puts M LuGre models i in parallel in order tocalculate the total friction torque Γ_(f)(t) as follows:

$\left\{ \begin{matrix}{{\Gamma_{f}(t)} = {{\sum\limits_{i = 1}^{M}\left( {{\sigma_{0,i}{z_{i}(t)}} + {\sigma_{1,i}{{\overset{.}{z}}_{i}(t)}}} \right)} + {\sigma_{2}{\omega(t)}}}} \\{{{\overset{.}{z}}_{i}(t)} = {{\omega(t)} - {\frac{❘{\omega(t)}❘}{g_{i}\left( {\omega(t)} \right)}{z_{i}(t)}}}} \\{{g_{i}(\omega)} = \frac{\Gamma_{c,i} + {\left( {\Gamma_{s,i} - \Gamma_{c,i}} \right)e^{- {(\frac{\omega(t)}{\omega_{s,i}})}^{2}}}}{\sigma_{0,i}}}\end{matrix} \right.$in which, for each model i:

-   -   σ_(0,i) is the stiffness of the spring blades representing the        friction roughnesses;    -   σ_(1,i) is the damping coefficient of the movement of the spring        blades;    -   σ₂ is the coefficient of viscous friction (it should be observed        that any need to make use of σ_(2,i) is avoided by        factorization);    -   z_(i) is the mean deformation of the blade;    -   Γ_(c,i) is the Coulomb torque;    -   Γ_(s,i) is the static friction torque;    -   ω_(s,i) is the Stribeck angular velocity; and    -   ω is the measured relative angular velocity of the motor.

The compensation structure, which is of the feedback type, is based onan estimate of the above model and it has the following form:

${g_{i}(\omega)} = {\frac{\Gamma_{c,i} + {\left( {\Gamma_{s,i} - \Gamma_{c,i}} \right)e^{- {(\frac{\omega}{\omega_{s,i}})}^{2}}}}{\sigma_{0,i}}.}$$\frac{d{\hat{z}}_{i}}{dt} = {\omega - {\frac{❘\omega ❘}{g_{i}(\omega)}{\hat{z}}_{i}} + {k_{s,i}{\varepsilon.}}}$ε = θ_(c) − θ_(r).${\hat{\Gamma}}_{f} = {{\sum\limits_{i = 1}^{M}\left( {{\sigma_{0,i}z_{i}} + {\sigma_{1,i}\frac{d{\hat{z}}_{i}}{dt}}} \right)} + {\sigma_{2}{\omega.}}}$in which:

-   -   ε is the error between the input setpoint or reference signal        (for position or speed) and the resulting output signal ⊖_(r)        (position or speed depending on the input setpoint);    -   the variables k_(s,i) are M in number and they are determined so        as to improve the robustness of the compensator in the face of        modifications to external conditions such as temperature or        changes in the system itself; and    -   the other parameters and variables are as mentioned above with        reference to the model.

Adjustment of the compensator includes the step of identifying all theparameters of each of the LuGre models i of the M-LuGre model and thestep of determining the variables k_(s,1), . . . , k_(s,i), . . . ,k_(s,M) that ensure good operation and that stabilize the compensator.

The parameters of the M-LuGre model are defined by using a stochasticoptimization algorithm, specifically a differential evolution algorithm.This algorithm is described in the document by P. Feyel et al., “LuGreFriction Model Identification and Compensator Tuning Using aDifferential Evolution Algorithm”, pages 85 to 91, 2013 IEEE Symposiumon Differential Evolution (SDE), 2013, IEEE. Preferably, the algorithmis used several times in succession in order to refine determination ofthe parameters of the model. How to determine the adjustment parametersfor the algorithm is itself known. It is possible to use the standardadjustment parameters for the differential evolution algorithm, such asthose mentioned in documents by R. Storn and K. Price, “DifferentialEvolution—A Simple and Efficient Heuristic for Global Optimization overContinuous Spaces”, pages 341-359, Journal of Global Optimization 11,Kluwer Academic Publisher, 1997.

In this example, the data for the mechanical system comprise itsspecifications and the results of tests or simulations performed on aservo control loop such as that shown in FIG. 5 for a system that isactuated by an actuator of the direct current (DC) motor type in orderto move a load that is assumed to be a pure inertial mass J. It can beunderstood that the servo control loop receives the setpoint ⊖_(c) asinput and that it delivers the output ⊖_(r). On the basis of thesetpoint ⊖_(c), there are determined both the product Js², which is thetorque needed to move the load through an angle θ_(c), and also theerror ε between the setpoint ⊖_(c) and the output ⊖_(r). The error ε isused in the corrector K(s) that delivers the command corresponding tothe Coulomb torque Γ_(c) from which there is subtracted the totalfriction Γ_(s)( ) as extracted from the output. The result of thissubtraction is divided by the product Js² in order to obtain the output⊖_(r).

Data is obtained for various different scenarios (corresponding todifferent conditions of use, in particular temperature, and to differentpower supply parameters for each of those conditions of use), and by wayof example it may comprise: parameters (voltage, current) of the powersupplied to the electrical actuator, speed of movement and force asdelivered by the actuator as a function of the power supply parameters,speed of movement and force that are to be obtained at the output fromthe mechanical system, . . . , which signals are sampled.

Thereafter, the differential evolution algorithm is used with thefollowing cost function, for a temperature Tc:

${{f_{T_{c}}(V)} = {\sum\limits_{k = 1}^{N_{m}}\left( {\frac{1}{N_{s,k}}{\sum\limits_{i = 1}^{N_{s,k}}\left( {{\theta_{r}\lbrack i\rbrack} - {{\hat{\theta}}_{r}\lbrack i\rbrack}} \right)^{2}}} \right)}},$in which:

-   -   Nm is the number of scenarios;    -   N_(s,k) is the number of samples; and    -   {circumflex over (⊖)}_(r) is the output from the model as        simulated with the following decision vector (σ_(0,i), σ_(1,i),        σ₂, Γ_(c,i), Γ_(s,i), ω_(s,i)), where i varies from 1 to M so        that a vector of (5M+1) parameters needs to be adjusted.

Once the parameters of the M-LuGre model have been determined for thevarious scenarios, it is possible to run a simulation on the basis ofthe servo control loop of FIG. 4 so as to determine both the command andthe sampled error for a given set of variables k_(s,1), . . . , k_(s,i),. . . , k_(s,M). For this purpose, the differential evolution algorithmis used once more, but with the following cost function:

${{f(V)} = {\sum\limits_{k = 1}^{N_{T}}\left( {\frac{1}{N_{s,k}}\begin{pmatrix}{{\sum\limits_{i = 1}^{N_{s,k}}\left( {{{Error}_{k}^{2}\lbrack i\rbrack} + {{\alpha\Gamma}_{c,k}^{2}\lbrack i\rbrack}} \right)} +} \\{\beta{\sum\limits_{i = 2}^{N_{s,k}}\left( {{\Gamma_{c,k}\lbrack i\rbrack} - {\Gamma_{c,k}\left\lbrack {i - 1} \right\rbrack}} \right)^{2}}}\end{pmatrix}} \right)}},$in which:

-   -   N_(T) is the number of scenarios tested;    -   N_(s,k) is the number of samples;    -   Γ_(c,k) is the command; and    -   α and β are positive real numbers selected to harmonize orders        of magnitude or to give greater importance to the error, to the        command, or to the variation of the command. For example: if α        and β are much less than 1, the error is amplified; if β is        zero, the derivative of the command is high.

FIGS. 3 and 4 show a feedback type compensator for a system that isactuated by a DC motor type actuator moving a load that is assumed to bea pure inertial mass J.

It can be understood that the compensator receives as input the setpoint⊖_(c) from which there are determined both the product Js² and the errorε between the setpoint ⊖_(c) and the output ⊖_(r). In this example, thecompensator has a compensation structure that is itself known (De Wit etal., “A new model for control of systems with friction”, pages 419 to425, No. 3, volume 40, IEEE Transactions on automatic control, 1995,IEEE). The error ε is used in the corrector K(s) and in the compensator{circumflex over (Γ)}_(f)( ) and their outputs are added to the productJs² in order to obtain the Coulomb torque Γ_(c) from which there issubtracted the total friction Γ_(s)( ) that results from the model. Theresult is divided by the product Js² in order to obtain the output⊖_(r).

This gives rise to a control relationship that incorporates frictioncompensation and that can be used to control the actuator in a mannerthat is effective, smooth, and stabilizing, since the structure of theoriginal LuGre compensation is conserved.

Naturally, the invention is not limited to the implementation described,but covers any variant coming within the field of the invention asdefined by the claims.

In particular, the method of the invention can be used for controllingan actuator of any kind of mechanical system. Such a mechanical systemis to produce accurate movement of a movable element, e.g. for aiming anoptical or chronic member in a given direction. A mechanical system maycomprise parts that are movable in rotation and/or in translation.

The term “LuGre model” is used to cover both the LuGre model as definedinitially, and also its derivatives such as the Dahl and Coulomb models.

The compensation structure may be of a type other than the feedbacktype, example it could be of feedforward type.

It is possible to use some other stochastic algorithm, or an algorithmother than a stochastic algorithm. For example, it is possible to usethe simplex algorithm, the pattern search algorithm, the subgradientmethod algorithm, the coordinate descent algorithm, . . . .

Using a global optimization algorithm without a gradient (referred to asa “derivative free optimization algorithm” in the literature) makes iteasier to perform the method, but is not essential.

The invention claimed is:
 1. A method of controlling an electricalactuator of a mechanical system having a plurality of nested zones ofcontact, the method comprising the steps of: acquiring data about themechanical system, which system includes a number of nested zones ofcontact; preparing a model of the system on the basis of said data andof a number of LuGre models equal to the number of nested zones ofcontact, and determining parameters of the model and also a compensationstructure for compensating friction in the nested zones of contact;including the compensation structure in a control relationship for theactuator; and controlling the actuator by means of the controlrelationship.
 2. The method according to claim 1, wherein the parametersof the model are determined by using a gradient free global optimizationalgorithm.
 3. The method according to claim 2, wherein the gradient freeglobal optimization algorithm is a stochastic algorithm.
 4. The methodaccording to claim 3, wherein the stochastic algorithm is a differentialevolution algorithm.
 5. The method according to claim 1, whereinmodelling parameters specific to each of the LuGre models i are:stiffness (σ_(0,i)) of blades representing friction; damping coefficient(σ_(1,i)) of movement of the blades; viscous friction coefficient (σ₂);Coulomb friction torque (Γ_(C,i)); static friction torque (Γ_(S,i));Stribeck angular velocity (ω_(S,i)); and global relative speed of theactuator (ω).
 6. The method according to claim 1, wherein thecompensation structure is of the feedback type.
 7. The method accordingto claim 1, wherein the compensation structure is of the feedforwardtype.